VISUALIZATION OF 3D GRAY-TONE IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 217

They are many different methods for mapping the shape of a 3D object
onto a 2D plane in descriptive geometry, computer graphics, drawing, and
painting. Details can be found in topics in each field. The two methods that
we discuss are sucient to visualize a 3D image.
In the normal (parallel) projection, lines running parallel in a 3D space
are drawn as parallel lines on a 2D plane. The length of a line segment and
the area of a face that is parallel to a 2D image plane is coinciding with real
values in the 3D space. The depth information is lost, however, and a drawn
result lacks in perspective.
In the perspective projection, a group of lines running parallel in a 3D space
and not parallel to an image plane are mapped to a group of lines intersecting
at a common intersection point on a 2D image plane. This intersection point
is called a vanishing point . The length of a line segment and the area of a face
on a 2D image plane can be changed according to the distance in the depth
direction, even if those lines and faces are parallel to an image plane. This
is called a perspective painting and was a basic method of European painting
after the Renaissance [Hockney01, Panofsky24].
Remark 7.5. In the wider sense, drawing perspective is the methodology for
feelings of the 3D world on a 2D plane. For example, details of a far object
are neglected without drawing, a distant view is drawn more bluish, a more
distant scene is drawn in the upper part of an image and the scene in the
foreground are in the lower area, a distant object is drawn vaguely and the
foreground view is drawn clearly. All of those are part of a perspective drawing.
More restrictive terms such as linear perspective may be used to represent the
perspective projection in the narrow sense as defined above.
Remark 7.6. In practical display equipment, a point on a display screen is
represented by its own coordinate system. The range of coordinate values is
definite. Therefore, the image plane coordinate system used in Section 7.4.1
has to be transformed once more into this coordinate system for every type of
display equipment. It is necessary to designate which part of an image plane
( H P in Fig. 7.6) in a virtual 3D space is displayed on a screen (window) on a
certain type of display equipment. This is called viewpoint transformation in
computer graphics (Fig. 7.7).
Remark 7.7. The whole of a 3D image that is to be visualized is not always
mapped onto an image plane H P in the mentioned imaginary 3D space. It
is sucient for the part that we want to observe to be rendered. An object
that is to be rendered is simplified for easiness in observation and to reduce
the computation time. Therefore, we frequently cut a 3D space by two planes
parallel to an image plane, and render only the part of the 3D space situated
between these two planes. In computer graphics this part of the 3D space
that is rendered is called the view volume . The view volume is a prismoid in
perspective projection and is in most cases a parallelepiped in the orthogonal
projection (Fig. 7.7).